On the characteristic polynomial of Cartan matrices and Chebyshev polynomials

Details On the characteristic polynomial of Cartan matrices and Chebyshev polynomials On the characteristic polynomial of Cartan matrices and Chebyshev polynomials

2011-10-30

arxiv.org/abs/1110.6620v2

Mathematics

Representation Theory

35 pages, minor corrections, one reference added

Scientific paper

We explore some interesting features of the characteristic polynomial of the Cartan matrix of a simple Lie algebra. The characteristic polynomial is closely related with the Chebyshev polynomials of first and second kind. In addition, we give explicit formulas for the characteristic polynomial of the Coxeter adjacency matrix, we compute the associated polynomials and use them to derive the Coxeter polynomial of the underlying graph. We determine the expression of the Coxeter and associated polynomials as a product of cyclotomic factors. We use this data to propose an algorithm for factoring Chebyshev polynomials over the integers. Finally, we prove an interesting formula which involves products of sines, the exponents, the Coxeter number and the determinant of the Cartan matrix.

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